Related papers: Constraints on hypothetical counterexamples to the…
The Casas-Alvero conjecture says that a degree $n$ complex univariate polynomial sharing a root with each of its derivative must have only one root. In this article we give three results. The first one, is that the number of possible…
Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been…
We present a proof of the Casas-Alvero conjecture, stating that if a complex polynomial has a root in common with each of its derivatives it must be a multiple of the power of some monomial.
The Casas--Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the…
The Casas-Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture…
We establish necessary and sufficient conditions for an arbitrary polynomial of degree $n$, especially with only real roots, to be trivial, i.e. to have the form a(x-b)^n. To do this, we derive new properties of polynomials and their roots.…
The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The…
In a first (theoretical) part of this paper, we prove a number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture, building on ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne that were recently…
We investigate necessary and sufficient conditions for an arbitrary polynomial of degree $n$ to be trivial, i.e. to have the form $a(z-b)^n$. These results are related to an open problem, conjectured in 2001 by E. Casas- Alvero. It says,…
According to Casas Alvero conjecture, if a one variable polynomial of degree $n$ over a field of characteristic 0 is not prime with each of the $n-1$ first derivees, then it is of the form $c (X-r)^{n}$. Let $p$ be a prime number and an…
The Casas-Alvero conjecture is about interpolation polynomials. There are some partial proofs of it, but there is not any proof in the general case.In this paper we propose three.
Based on Casas-Alvero conjecture \textit{[J. Algebra, 2001]} we formulate the following conjecture.\\ \textbf{C*-algebraic Casas-Alvero Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra, $n\in \mathbb{N}$ and let $P(z) =…
The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\_i(f)$ is a power of a linear polynomial. Let…
The Casas-Alvero conjecture states that if $f(X)$ is a monic univariate polynomial of degree $d$ over a characteristic $0$ field $\mathbb{K}$ such that $\gcd(f, f_{i})$ is non-trivial for each $i=1, \dots, d-1$, then $f(X)=(X-\alpha)^d$ for…
An outstanding conjecture on roots of Ehrhart polynomials says that all roots $\alpha$ of the Ehrhart polynomial of an integral convex polytope of dimension $d$ satisfy $-d \leq \Re(\alpha) \leq d-1$. In this paper, we suggest some…
We conjecture that the roots of a degree-n univariate complex polynomial are located in a union of n-1 annuli, each of which is centered at a root of the derivative and whose radii depend on higher derivatives. We prove the conjecture for…
We study varieties defined by parameterizing polynomials of derivatives through a computational algebro-geometric approach, especially relying on Combinatorial Nullstellensatz and Noether normalization. We establish that these polynomials…
In this paper we first prove that a simple root of a polynomial satisfies the Sendov's conjecture. As the multiple roots trivially satisfy the Sendov's conjecture we conclude that the Sendov's conjecture holds true.
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger…
For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers $pos$ of positive and $neg$ of negative roots (counted with multiplicity) are majorized…